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Chapter 1
A Mathematical Theory of Climate Sensitivity or,
How to Deal With Both Anthropogenic Forcing and Natural Variability?
Michael Ghil
Ecole Normale Superieure, 75005 Paris, FRANCE, andUniversity of California, Los Angeles, CA 90095, USA
Recent estimates of climate evolution over the coming century still differ by several degrees. Thisuncertainty motivates the work presented here. There are two basic approaches to apprehend thecomplexity of climate change: deterministically nonlinear and stochastically linear, i.e. the Lorenzand the Hasselmann approach. The grand unification of these two approaches relies on the theoryof random dynamical systems. We apply this theory to study the random attractors of nonlinear,stochastically perturbed climate models. Doing so allows one to examine the interaction of internalclimate variability with the forcing, whether natural or anthropogenic, and to take into account theclimate system’s non-equilibrium behavior in determining climate sensitivity.
This non-equilibrium behavior is due to a combination of nonlinear and random effects. We givehere a unified treatment of such effects from the point of view of the theory of dynamical systemsand of their bifurcations. Energy balance models are used to illustrate multiple equilibria, whilemulti-decadal oscillations in the thermohaline circulation illustrate the transition from steady statesto periodic behavior. Random effects are introduced in the setting of random dynamical systems,which permit a unified treatment of both nonlinearity and stochasticity. The combined treatment ofnonlinear and random effects is applied to a stochastically perturbed version of the classical Lorenzconvection model.
Climate sensitivity is then defined mathematically as the derivative of an appropriate functionalor other function of the systems state with respect to the bifurcation parameter. This definition isillustrated by using numerical results for a model of the El Nino–Southern Oscillation.
1.1. Introduction
The global climate system is composed of a num-
ber of subsystems — atmosphere, biosphere,
cryosphere, hydrosphere and lithosphere — each
of which has distinct characteristic times, from
days and weeks to centuries and millennia. Each
subsystem, moreover, has its own internal vari-
ability, all other things being constant, over a
fairly broad range of time scales. These ranges
overlap between one subsystem and another.
The interactions between the subsystems thus
give rise to climate variability on all time scales.
We outline here the rudiments of the way
in which dynamical systems theory provides an
understanding of this vast range of variability.
Such an understanding proceeds through the
study of successively more complex patterns of
behavior. These spatio-temporal patterns are
studied within narrower ranges of time scales,
such as intraseasonal, interannual, interdecadal
and multi-millennial. The main results of dy-
namical systems theory that have demonstrated
their importance for the study of climate vari-
ability involve bifurcation theory and the ergodic
theory of dynamical systems. More recently, the
∗additional E-address: [email protected].
1
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2 M. Ghil
theory of random dynamical systems has made
substantial contributions as well.
In the next section, we describe the cli-
mate systems dominant balance between incom-
ing solar radiation, dominated by short waves,
and outgoing terrestrial radiation, dominated
by long waves. This balance is consistent with
the existence of multiple equilibria of surface
temperatures.
Such multiple equilibria are also present for
other balances of climatic actions and reactions.
Thus, on the intraseasonal time scale, the ther-
mal driving of the mid-latitude westerly winds
is countered by surface friction and mountain
drag. Multiple equilibria typically arise from
saddle-node bifurcations of the governing equa-
tions. Transitions from one equilibrium to an-
other may result from small and random pushes
— a typical case of minute causes having large
effects in the long term.
In Sec. 1.3, we sketch the oceans overturning
circulation between cold regions, where water is
heavier and sinks, and warm regions, where it is
lighter and rises. The effect of temperature on
the water masses density and, hence, motion is
in competition with the effect of salinity: density
increases, through evaporation and brine forma-
tion, compete further with decreases in salinity
and, hence, density through precipitation and
river run-off. These competing effects can also
give rise to two distinct equilibria.
In the present-day oceans, a thermohaline
circulation prevails, in which the temperature
effects dominate. In the remote past, about 50
Myr ago, a halothermal circulation may have
obtained, with salinity effects dominating. In a
simplified mathematical setting, these two equi-
libria arise by a pitchfork bifurcation that breaks
the problems mirror symmetry. On shorter time
scales, of decades-to-millennia, oscillations of in-
tensity and spatial pattern in the thermohaline
circulation seem to be the dominant mode of
variability. We show how interdecadal oscilla-
tions in the oceans circulation arise by Hopf
bifurcation.
In Sec. 1.4, we address the way that faster
processes, modeled as random effects, can inter-
act with the slower, nonlinear ones. The com-
bined treatment of the nonlinear and stochastic
processes can reveal amazingly fine structure in
the climate systems behavior, but also — and
rather surprisingly — add robustness and pre-
dictability to the results.
In Sec. 1.5, we discuss the way that climate
sensitivity can be defined in the stochastic vs.
the deterministic context. Concluding remarks
follow in Sec. 1.6.
1.2. Energy-Balance Models and the
Modeling Hierarchy
The concepts and methods of the theory of de-
terministic dynamical systems (Andronov and
Pontryagin, 1937; Arnol’d, 1983; Guckenheimer
and Holmes, 1983) have been applied first
to simple models of atmospheric and oceanic
flows, starting about fifty years ago (Lorenz,
1963; Stommel, 1961). More powerful comput-
ers now allow their application to fairly realistic
and detailed models of the atmosphere, ocean,
and the coupled atmosphereocean system. We
start therefore by presenting such a hierarchy
of models.
This presentation is interwoven with that of
the successive bifurcations that lead from simple
to more complex solution behavior for each cli-
mate model. Useful tools for comparing model
behavior across the hierarchy and with observa-
tions are provided by ergodic theory (Eckmann
and Ruelle, 1985; Ghil et al., 2008a). Among
these, advanced methods for the analysis and
prediction of uni- and multivariate time series
play an important role.
1.2.1. Radiation balance and energy-
balance models (EBMs)
The concept of a modeling hierarchy in cli-
mate dynamics was introduced by Schneider
and Dickinson (1974); it is discussed in greater
detail by Ghil and Robertson (2000) and by
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A Mathematical Theory of Climate Sensitivity 3
Dijkstra and Ghil (2005). At present, the best-
developed hierarchy is for atmospheric mod-
els. These models were originally developed for
weather simulation and prediction on the time
scale of hours to days. Currently they serve in
a stand-alone mode or coupled to oceanic and
other models to address climate variability on
all time scales.
The first rung of the modeling hierarchy for
the atmosphere is formed by zero-dimensional
(0-D) models, where the number of dimensions,
from zero to three, refers to the number of in-
dependent space variables used to describe the
model domain, i.e. to physical-space dimensions.
Such 0-D models essentially attempt to follow
the evolution of global surface-air temperature
as a result of changes in global radiative balance:
cdT
dt= Ri −Ro, (1.1a)
Ri = µQ0{1− α(T )}, (1.1b)
Ro = σm(T )(T )4. (1.1c)
Here Ri and Ro are incoming solar radiation
and outgoing terrestrial radiation. The heat ca-
pacity c is that of the global atmosphere, plus
that of the global ocean or some fraction thereof,
depending on the time scale of interest: one
might only include in c the ocean mixed layer
when interested in subannual time scales but the
entire ocean when studying paleoclimate.
The rate of change of T with time t is given
by dT /dt, while Q0 is the solar radiation re-
ceived at the top of the atmosphere, σ is the Ste-
fanBoltzmann constant, and m is an insolation
parameter, equal to unity for present-day con-
ditions. To have a closed, self-consistent model,
the planetary reflectivity or albedo α and gray-
ness factor m have to be expressed as functions
of T ; m = 1 for a perfectly black body and
0 < m < 1 for a grey body like planet Earth.
There are two kinds of one-dimensional (1-
D) atmospheric models, for which the single spa-
tial variable is latitude or height, respectively.
The former are so-called energy-balance mod-
els (EBMs), which consider the generalization of
the model (2.1) for the evolution of surface-air
temperature T = T (x, t), say,
c(x)∂T
∂t= Ri −Ro +D. (1.2)
Here the terms on the right-hand side can be
functions of the meridional coordinate x (lat-
itude, co-latitude, or sine of latitude), as well
as of time t and temperature T . The horizontal
heat-flux term D expresses heat exchange be-
tween latitude belts; it typically contains first
and second partial derivatives of T with respect
to x. Hence the rate of change of local tempera-
ture T with respect to time also becomes a par-
tial derivative, ∂T /∂t. Such models were intro-
duced independently by Budyko (1969) and by
Sellers (1969).
The first striking results of theoretical cli-
mate dynamics were obtained in showing that
Eq. (1.2) could have two stable steady-state so-
lutions, depending on the value of the insolation
parameter µ, cf. Eq. (1.1b). This multiplicity of
stable steady states, or physically possible cli-
mates of our planet, can be explained, in its sim-
plest form, in the 0-D model given by Eq. (1.1).
The physical explanation resides in the fact
that — for a fairly broad range of µ-values
around µ = 1.0 — the curves for Ri and Ro as a
function of T intersect in 3 points. One of these
points corresponds to the present climate (high-
est T -value), and another one to an ice-covered
planet (lowest T -value); both of these are sta-
ble, while the third one (intermediate T -value)
is unstable. To obtain this result, it suffices to
make two assumptions: (i) that α = α(T ) is a
piecewise-linear function of T , with high albedo
at low temperature, due to the presence of snow
and ice, and low albedo at high T , due to their
absence; and (ii) that m = m(T ) is a smooth,
increasing function of T that attempts to cap-
ture in its simplest from the greenhouse effect of
trace gases and water vapor.
The bifurcation diagram of a 1-D EBM, like
the one of Eq. (1.2), is shown in Fig. 1.1. It
displays the model’s mean temperature T as a
function of the fractional change µ in the inso-
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4 M. Ghil
lation Q = Q(x) at the top of the atmosphere.
The S-shaped curve in the figure arises from two
back-to-back saddle-node bifurcations.
Fig. 1.1. Bifurcation diagram for the solutions of an
energy-balance model (EBM), showing the global-mean
temperature T vs. the fractional change µ of insolationat the top of the atmosphere. The arrows pointing up
and down at about µ = 1.4 indicate the stability of
the branches: towards a given branch if it is stable andaway if it is unstable. The other arrows show the hys-
teresis cycle that global temperatures would have to un-dergo for transition from the upper stable branch to the
lower one and back. The angle γ gives the measure of
the present climates sensitivity to changes in insolation.[After Ghil and Childress (1987) with permission from
Springer-Verlag.]
The normal form of the first one is
X = µ−X2. (1.3)
Here X stands for a suitably normalized form
of T and X is the rate of change of X, while µ
is a parameter that measures the stress on the
system, in particular a normalized form of the
insolation parameter.
The uppermost branch corresponds to the
steady-state solution X = +µ1/2 of Eq. (1.3)
and it is stable. This branch matches rather well
Earth’s present-day climate for µ = 1.0; more
precisely the steady-state solution T = T (x;µ)
of the full 1-D EBM (not shown) matches closely
the annual mean temperature profile from in-
strumental data over the last century.
The intermediate branch starts out at the
left as the second solution, X = −µ1/2, of
Eq. (1.3) and it is unstable. It blends smoothly
into the upper branch of a coordinate-shifted
and mirror-reflected version of Eq. (1.3), say
X = µ−X2. (1.4)
This branch, X = X0+(µ0−µ)1/2, is also un-
stable. Finally, the lowermost branch in Fig. 1.1
is the second steady-state solution of Eq. (1.4),
X = X0(µ0 − µ)1/2, and it is stable, like the
uppermost branch. The lowermost branch cor-
responds to an ice-covered planet at the same
distance from the Sun as Earth.
The fact that the upper-left bifurcation point
(µc, Tc) in Fig. 1.1 is so close to present-
day insolation values created great concern in
the climate dynamics community in the mid-
1970s, when these results were obtained. In-
deed, much more detailed computations (see be-
low) confirmed that a reduction of about 2–5%
of insolation values would suffice to precipitate
Earth into a deep freeze. The great distance of
the lower-right bifurcation point (µd, Td) from
present-day insolation values, on the other hand,
suggests that one would have to nearly double
atmospheric opacity, say, for the Earth’s climate
to jump back to more comfortable temperatures.
The results here follow Ghil (1976). Held and
Suarez (1974) and North North (1975) obtained
similar results, and a detailed comparison be-
tween EBMs appears in Chapter 10 of Ghil and
Childress (1987).
1.2.2. Other atmospheric processes
and models
The 1-D atmospheric models in which the de-
tails of radiative equilibrium are investigated
with respect to a height coordinate z (geo-
metric height, pressure, etc.) are often called
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A Mathematical Theory of Climate Sensitivity 5
radiative-convective models (Ramanathan and
Coakley, 1978). This name emphasizes the key
role that convection plays in vertical heat trans-
fer. While these models preceded historically
EBMs as rungs on the modeling hierarchy, it
was only recently shown that they, too, could
exhibit multiple equilibria (Li et al., 1997). The
word (stable) equilibrium, here and in the rest
of this article, refers simply to a (stable) steady
state of the model, rather than a to true ther-
modynamic equilibrium.
Two-dimensional (2-D) atmospheric models
are also of two kinds, according to the third
space coordinate that is not explicitly included.
Models that resolve explicitly two horizontal co-
ordinates, on the sphere or on a plane tangent to
it, tend to emphasize the study of the dynamics
of large-scale atmospheric motions. They often
have a single layer or two. Those that resolve ex-
plicitly a meridional coordinate and height are
essentially combinations of EBMs and radiative-
convective models and emphasize therewith the
thermodynamic state of the system, rather than
its dynamics.
Yet another class of horizontal 2-D models
is the extension of EBMs to resolve zonal, as
well as meridional surface features, in particu-
lar land-sea contrasts. We shall see in Sec. 1.3.2
how such a 2-D EBM is used, when coupled to
an oceanic model.
Schneider and Dickinson (1974) and Ghil
and Robertson (2000) discuss additional types of
1-D and 2-D atmospheric models and give refer-
ences to these and to the types discussed above,
along with some of their main applications. Fi-
nally, to encompass and resolve the main at-
mospheric phenomena with respect to all three
spatial coordinates, general circulation models
(GCMs) occupy the pinnacle of the modeling
hierarchy.
The dependence of mean zonal temperature
on the insolation parameter µ (the normalized
“solar constant”) — as obtained for 1-D EBMs
and shown in Fig. 1.1 here — was confirmed, to
the extent possible, by using a simplified GCM,
coupled to a swamp ocean model. More pre-
cisely, forward integrations with a GCM cannot
confirm the presence of the intermediate, unsta-
ble branch. Nor was it possible in the mid-70s,
when this numerical experiment was done, to
reach the deep-freeze stable branch, because of
the GCMs computational limitations. But the
parabolic shape of the upper, present-daylike
branch near the upper-left bifurcation point in
our figure, cf. Eq. (1.3), was well supported by
the GCM simulations.
Ghil and Robertson (2000) also describe the
separate hierarchies that have grown over the
last quarter-century in modeling the ocean and
the coupled oceanatmosphere system. More re-
cently, an overarching hierarchy of earth-system
models that encompass all the subsystems of
interest, atmosphere, biosphere, cryosphere, hy-
drosphere and lithosphere has been developing.
Eventually, the partial results about each sub-
systems variability, outlined in this section and
the next one, will have to be verified from one
rung to the next of the Earth-system modeling
hierarchy.
1.3. Oscillations in the Oceans’
Thermohaline Circulation
1.3.1. Theory and simple models
Historically, the thermohaline circulation
(THC) was first among the climate systems
major processes to be studied using a very sim-
ple mathematical model. Stommel (1961) for-
mulated a two-box model and showed that it
possessed multiple equilibria.
A sketch of the Atlantic Oceans THC
and its interactions with the atmosphere and
cryosphere on long time scales is shown in
Fig. 1.2. These interactions can lead to climate
oscillations with multi-millennial periods, such
as the Heinrich events, and are summarized in
the figures caption. An equally schematic view
of the global THC is provided by the widely
known “conveyor belt” diagram. The latter dia-
gram captures greater horizontal, 2-D detail but
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6 M. Ghil
it does not commonly include the THCs inter-
actions with water in both its gaseous and solid
phases, which our Fig. 1.2 here does include.
Fig. 1.2. Diagram of an Atlantic meridional cross sec-
tion from North Pole (NP) to South Pole (SP), showingmechanisms likely to affect the thermohaline circulation
(THC) on various time-scales. Changes in the radiation
balance Rin − Rout are due, at least in part, to changesin extent of Northern Hemisphere (NH) snow and ice
cover V , and to how these changes affect the global tem-
perature T ; the extent of Southern Hemisphere ice isassumed constant, to a first approximation. The change
in hydrologic cycle expressed in the terms Prain − Pevap
for the ocean and Psnow − Pabl for the snow and ice isdue to changes in ocean temperature. Deep-water forma-
tion in the North Atlantic Subpolar Sea (North Atlantic
Deep Water: NADW) is affected by changes in ice volumeand extent, and regulates the intensity C of the THC;changes in Antarctic Bottom Water (AABW) formation
are neglected in this approximation. The THC intensityC in turn affects the systems temperature, and is also
affected by it. [After Ghil et al. (1987) with permissionfrom Springer-Verlag.]
Basically, the THC is due to denser wa-
ter sinking, lighter water rising, and water-
mass continuity closing the circuit through near-
horizontal flow between the areas of rising and
sinking. The effects of temperature and salinity
on the ocean waters density, ρ = ρ(T, S), op-
pose each other: the density ρ decreases as T
increases and it increases as S increases. It is
these two effects that give the thermohaline cir-
culation its name, from the Greek words for T
and S. In high latitudes, ρ increases as the wa-
ter loses heat to the air above and, if sea ice is
formed, as the water underneath is enriched in
brine. In low latitudes, ρ increases due to evap-
oration but decreases due to sensible heat flux
into the ocean.
For the present climate, the temperature ef-
fect is stronger than the salinity effect, and
ocean water is observed to sink in certain areas
of the high-latitude North Atlantic and South-
ern Ocean — with very few and limited areas of
deep-water formation elsewhere — and to rise
everywhere else. Thus, in a thermohaline regime,
T is more important than S and hence comes
before it. During some remote geological times,
deep water may have formed in the global ocean
near the equator; such an overturning circula-
tion of opposite sign to that prevailing today
has been dubbed halothermal, S before T . The
quantification of the relative effects of T and S
on the oceanic water masses buoyancy in high
and low latitudes is far from complete, especially
for paleocirculations; the association of the lat-
ter with salinity effects that exceed the thermal
ones (Kennett and Stott, 1991) is thus rather
tentative.
To study the reversal of the abyssal circu-
lation, due to the opposite effects of T and S,
Stommel considered a two-box model, with two
pipes connecting the two boxes. He showed that
the system of two nonlinear, coupled ordinary
differential equations that govern the temper-
ature and salinity differences between the two
well-mixed boxes has two stable steady-state
solutions; these two steady states are distin-
guished by the direction of flow in the upper and
lower pipe. Stommel’s paper was primarily con-
cerned with distinct local convection regimes,
and hence vertical stratifications, in the North
Atlantic and the Mediterranean or the Red Sea,
say. Today, we mainly think of one box as rep-
resenting the low latitudes and the other one
the high latitudes in the global THC (Marotzke,
2000).
The next step in the hierarchical modeling of
the THC is that of 2-D meridional plane models,
in which the temperature and salinity fields are
governed by coupled nonlinear partial differen-
tial equations with two independent space vari-
ables, latitude and depth, say. Given boundary
conditions for such a model that are symmetric
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A Mathematical Theory of Climate Sensitivity 7
about the Equator, as the equations themselves
are, one expects a symmetric solution, in which
water either sinks near the poles and rises ev-
erywhere else (thermohaline) or sinks near the
Equator and rises everywhere else (halother-
mal). These two symmetric solutions would cor-
respond to the two equilibria of Stommel’s box
model of 1961.
In fact, symmetry breaking can occur, leading
gradually from a symmetric two-cell circulation
to an antisymmetric one-cell circulation. In be-
tween, all degrees of dominance of one cell over
the other are possible. A situation lying some-
where between the two seems to resemble most
closely the meridional overturning diagram of
the Atlantic Ocean in Fig. 1.2.
This symmetry breaking can be described by
a pitchfork bifurcation:
X = µ−X3. (1.5)
Here X stands for the amount of asymmetry in
the solution, so that X = 0 is the symmetric
branch, and µ is a parameter that measures the
stress on the system, in particular a normalized
form of the buoyancy flux at the surface. For
µ < 0 the symmetric branch is stable, while for
µ > 0 the two branches X = ±µ1/2 inherit its
stability.
In the 2-D THC problem, the left cell dom-
inates on one branch, while the right cell dom-
inates on the other: for a given value of µ, the
two stable steady-state solutions — on the {X =
+µ1/2} branch and on the {X = −µ1/2} branch
— are mirror images of each other. The idealized
THC in Fig. 1.2, with the North Atlantic Deep
Water extending to the Southern Oceans polar
front, corresponds to one of these two branches.
In theory, therefore, a mirror-image circulation,
with the Antarctic Bottom Water extending
to the North Atlantics polar front, is equally
possible.
1.3.2. Bifurcation diagrams for GCMs
Bryan (1986) was the first to document transi-
tion from a two-cell to a one-cell circulation in a
simplified ocean GCM with idealized, symmet-
ric forcing. Results of coupled oceanatmosphere
GCMs, however, have led to questions about the
realism of more than one stable THC equilib-
rium. The situation with respect to the THCs
pitchfork bifurcation (1.5) is thus subtler than
it was with respect to Fig. 1.1 for radiative equi-
librium. In the previous section, atmospheric
GCMs confirmed essentially the EBM results;
the results obtained in climbing the rungs of the
modeling hierarchy for the THC are still in need
of further clarification.
Internal variability of the THC with smaller
and more regular excursions than the huge and
totally irregular jumps associated with bistabil-
ity was studied intensively in the late 1980s and
the 1990s. These studies placed themselves on
various rungs of the modeling hierarchy, from
box models through 2-D models and all the way
to ocean GCMs. A summary of the different
kinds of oscillatory variability found in the latter
appears in Table 1.1. Such oscillatory behavior
seems to match more closely the instrumentally
recorded THC variability, as well as the paleo-
climatic records for the recent geological past,
than bistability.
The (multi)millennial oscillations interact
with variability in the surface features and pro-
cesses shown in Fig. 1.2. Chen and Ghil (1996),
in particular, studied some of the interactions
between atmospheric processes and the THC.
They used a so-called hybrid coupled model,
namely a (horizontally) 2-D EBM, coupled to
a rectangular-box version of the North At-
lantic rendered by a low-resolution ocean GCM.
This hybrid models regime diagram is shown in
Fig. 1.3(a). A steady state is stable for high val-
ues of the coupling parameter or of the EBMs
diffusion parameter d. Interdecadal oscillations
with a period of 40–50 years are self-sustained
and stable for low values of these parameters.
The self-sustained THC oscillations in ques-
tion are characterized by a pair of vortices of
opposite sign that grow and decay in quadra-
ture with each other in the oceans upper layers.
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Table 1.1. Oscillations in the oceans’ thermohaline circulation.
Time scale Phenomena Mechanisma
Decadal · Local migration of surface anomalies in the
northwest corner of the ocean basin
· Localized surface density anomalies due to
surface coupling· Gyre advection in mid-latitudes · Gyre advection
Centennial Loop-type, meridional circulation Conveyor-belt advection of density anomalies
Millennial Relaxation oscillation, with “flushes” and su-perimposed decadal fluctuations
Bottom water warming, due to strong brakingeffect of salinity forcing
a Full references to these mechanisms are given in Ghil (1994).a) Regime diagram
b) Bifurcation diagram
a) Regime diagram
b) Bifurcation diagram
a) Regime diagram b) Bifurcation diagram
Fig. 1.3. Dependence of THC solutions on two parameters in a hybrid coupled model; the two parameters are theatmosphere-ocean coupling coefficient λao and the atmospheric thermal diffusion coefficient d. (a) Schematic regime
diagram. The full circles stand for the models stable steady states, the open circles for stable limit cycles, and the solid
curve is the estimated neutral stability curve between the former and the latter. (b) Hopf bifurcation curve at fixed d= 1.0 and varying λao; this curve was obtained by fitting a parabola to the models numerical-simulation results, shown
as full and open circles. [From Chen and Ghil (1996) with permission from the American Meteorological Society.]
Their centers follow each other anti-clockwise
through the northwestern quadrant of the mod-
els rectangular domain. Both the period and the
spatio-temporal characteristics of the oscillation
are thus rather similar to those seen in a fully
coupled GCM with realistic geometry. The tran-
sition from a stable equilibrium to a stable limit
cycle, via Hopf bifurcation, in this hybrid cou-
pled model, is shown in Fig. 1.3(b).
1.4. Randomness and Nonlinearity
1.4.1. What to expect
The geometric (Arnol’d, 1983; Guckenheimer
and Holmes, 1983) and the ergodic (Eckmann
and Ruelle, 1985) theory of dynamical systems
represent significant achievements of the 20th
century. The foundations of the stochastic calcu-
lus in its second half (Doob, 1953) also led to the
birth of a rigorous theory of time-dependent ran-
dom phenomena. Historically, theoretical devel-
opments in climate dynamics have been largely
motivated by these two complementary ap-
proaches, based on the work of E. N. Lorenz
and that of K. Hasselmann (Lorenz, 1963; Has-
selmann, 1976), respectively.
It now seems clear that these two approaches
complement, rather than exclude each other.
Incomplete knowledge of small-, subgrid-scale
processes, as well as computational limitations
will always require one to account for these
processes in a stochastic way. As a result of
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A Mathematical Theory of Climate Sensitivity 9
sensitive dependence on initial data and on pa-
rameters, numerical weather forecasts, as well
as climate projections are both expressed these
days in probabilistic terms. In addition to the
intrinsic challenge of addressing the nonlinear-
ity along with the stochasticity of climatic pro-
cesses, it is thus more convenient — and becom-
ing more and more necessary — to rely on a
model’s (or set of models’) probability density
function (PDF) rather than on its individual,
point-wise simulations or predictions.
We summarize here results on the surpris-
ingly complex statistical structure that charac-
terizes stochastic nonlinear systems. This com-
plex structure does provide meaningful phys-
ical information that is not described by the
PDF alone; it lives on a random attractor,
which extends the concepts of a strange attractor
and of the invariant measure that is supported
by it, from the deterministic to the stochastic
framework.
1.4.2. What one finds
On the road to including random effects, one
needs to realize first that the climate system —
as well as any of its subsystems, and on any
time scale — is not closed: it exchanges en-
ergy, mass and momentum with its surround-
ings, whether other subsystems or the inter-
planetary space and the solid earth. The typi-
cal applications of dynamical systems theory to
climate variability so far have only taken into
account exchanges that are constant in time,
thus keeping the model — whether governed by
ordinary, partial or other differential equations
— autonomous; i.e., the models had coefficients
and forcings that were constant in time.
Succinctly, one can write such a system as
X = f(X;µ), (1.6)
where X now may stand for any state vector or
climate field, while f is a smooth function of X
and of the vector of parameters µ, but does not
depend explicitly on time. This characteristic of
being autonomous greatly facilitated the analy-
sis of model solutions’ properties. For instance,
two distinct trajectories, X1(t) and X2(t), of a
well-behaved, smooth autonomous system can-
not pass through the same point in phase space,
which helps describe the systems phase portrait.
So does the fact that we only need to consider
the behavior of solutions X(t) as we let time
t tend to +∞: the resulting sets of points are
— possibly multiple — equilibria, periodic so-
lutions, and chaotic sets. In the language of
dynamical systems theory, these are called, re-
spectively: fixed points, limit cycles, and strange
attractors.
We know only too well, however, that the
seasonal cycle plays a key role in climate vari-
ability on many time scales, while orbital forc-
ing is crucial on the Quaternary time scales
of many millennia, and now anthropogenic forc-
ing is of utmost importance on interdecadal
time scales. How can one take into account such
time-dependent forcings, and analyze the non-
autonomous systems, written succinctly as
X = f(X, t;µ), (1.7)
to which they give rise? In Eq. (1.7), the depen-
dence of f on t may be periodic, f(X, t + P ) =
f(X, t), as in various El Nino–Southern Oscilla-
tion (ENSO) models, where the period P = 12
months, or monotone, f(X, t + τ) ≥ f(X, t) for
τ ≥ 0, as in studying scenarios of anthropogenic
climate forcing.
To illustrate the fundamental character of
the distinction between an autonomous system
like (1.6) and a non-autonomous one like (1.7),
consider the simple scalar version of these two
equations:
X = −βX, (1.8)
and
X = −βX + γt, (1.9)
respectively. We assume that both systems are
dissipative, i.e. β > 0, and that the forcing is
monotone increasing, γ ≥ 0, as would be the
case for anthropogenic forcing in the industrial
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10 M. Ghil
era. Lorenz (1963) pointed out the key role of
dissipativity in giving rise to strange, but at-
tracting solution behavior, while Ghil and Chil-
dress (1987) emphasized its importance and per-
vasive character in climate dynamics. Clearly
the only attractor for the solutions of Eq. (1.8),
given any initial point X(0) = X0, is the fixed
point X = 0, attained as t→ +∞.
For the non-autonomous case of Eq. (1.9),
though, this forward-in-time approach yields
blow-up as t → +∞, for any initial point. To
make sense of what happens in the case of time-
dependent forcing, one introduces instead the
pullback approach, in which solutions are allowed
to still depend on the time t at which we observe
them, but also on a time s from which the solu-
tion is started, X(s) = X0; presumably s � t.
With this little change of approach, one can eas-
ily verify that
|X(s, t;X0)−A(t)| → 0 as s→ −∞,(1.10)
for all t and X0, where
A(t) =γ(t− 1/β)
β. (1.11)
We thus obtain, in this pullback sense, the in-
tuitively obvious result that the solutions, if
started far enough in the past, all approach the
attractor set A(ω), which has a linear growth in
time and thus follows the linear forcing.
Let us return now to the more general, non-
linear case of Eq. (1.7) and add not only de-
terministic time dependence f(X, t), but also
random forcing (Ledrappier and Young, 1988;
Arnold, 1998),
dX = f(X, t)dt+ g(X)dη, (1.12)
where η = η(t;ω) represents a Wiener process
— and dη(t) is commonly referred to as “white
noise — while ω labels the particular realiza-
tion of this random process. The case g(X) =
const. is the case of additive noise, while in the
case of ∂g(X)/∂X 6= 0 we speak of multiplica-
tive noise. The distinction between dt and dω in
Eq. (1.12) is necessary since, roughly speaking
and following Einstein’s treatment of Brownian
motion (Einstein, 1905), it is the variance of a
Wiener process that is proportional to time and
thus dη ∼ (dt)1/2.
In the case of random forcing, we can illus-
trate the concepts introduced by the simple ex-
ample of Eqs. (1.10, 1.11) above by the random
attractor A(ω) (yellow band) of Fig. 1.4. In the
figure, dη(t;ω) = θ(t)ω is the random process
that drives the system (solid black line) and the
pullback attraction is depicted by the flow of an
arbitrary set B from “pullback times t = −τ2and t = −τ1 onto the attractor (heavy blue
arrows).
{!( )"}xt X{"}xX
!( )"t
A( )" #( ,") t A( )=" t!( )"A( )
$"
Pullback attraction to A( )"
2!(%& )"!(%& )"1
!(%& )")B(!(%& )")B( 1
2
Fig. 1.4. Schematic diagram of a random attractor
A(ω) and of the pullback attraction to it; here ω labelsthe particular realization of the random process θ(t)ω
that drives the system. We illustrate the evolution intime t of the random process θ(t)ω (solid black line at
the bottom); the random attractor A(ω) itself (yellowband in the middle) with the “snapshots” A(ω) = A(t =0;ω) and A(t;ω) (the two vertical sections, heavy solid);
and the flow of an arbitrary set B from pullback times
t = −τ2 and t = −τ1 onto the attractor (heavy bluearrows). [After Ghil et al. (2008a) with permission from
Elsevier.]
More explicitly, we show in Fig. 1.5 four
snapshots {Aj(ω) = A(ω; t = tj) : j = 1, 2, 3, 4}that correspond to the vertical cross-sections
(heavy solid) in the attractor of Fig. 1.4; a
short video, from which these snapshots are
taken, appears as Supplementary Information in
Chekroun et al. (2011b). These snapshots were
calculated for the random attractor A(t;ω) of
a stochastically perturbed Lorenz system, given
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A Mathematical Theory of Climate Sensitivity 11
by
dX = Pr(Y −X)dt+ σXdη, (1.13a)
dX = (rX − Y −XZ)dt+ σY dη (1.13b)
dX = (−bZ +XY )dt+ σZdη. (1.13c)
Fig. 1.5. Four snapshots of the stochastically perturbed
Lorenz (1963) model’s random attractor A(ω) and the
invariant measure ν(ω) supported on it. The parame-ter values are the classical ones — normalized Rayleigh
number r = 28, Prandtl number Pr = 10, and normal-
ized wave number b = 8/3 — while the noise intensityis σ = 0.5 and the time step is δt = 5 · 103. The color
bar used is on a log-scale and quantifies the probability
to end up in a particular region of phase space; shownis a projection of the 3-D phase space (X,Y, Z) onto
the (X,Z)-plane. Notice the complex, interlaced filament
structures between highly (yellow) and moderately (red)populated regions. The time interval ∆t between two suc-cessive snapshots — moving from left to right and fromtop to bottom — is ∆t = 0.0875. Note that the sup-port of the invariant measure ν(t;ω) may change quite
abruptly, from time t to time t+∆t; see the related shortvideo in Chekroun et al. (2011b). Weakly populated re-
gions cover an important part of the random attractorand are, in turn, entangled with regions that have near-zero probability (black). [After Chekroun et al. (2011b)with permission from Elsevier.]
The parameters r, Pr and b in Eqs. (1.13)
have the usual meanings for 2-D thermal con-
vection: r = R/Rc is the Rayleigh number R
normalized by its critical value Rc at the onset
of convection, Pr is the Prandtl number, and b
is a normalized wave number for the most unsta-
ble wave at the onset of convection. The noise in
this case is multiplicative: its intensity σ = 0.5
is multiplied in each one of the three coupled,
nonlinear equations above by the corresponding
variable X,Y or Z.
To be precise, what is plotted in Fig. 1.5, and
in the associated video, is the density of the in-
variant measure ν(ω) supported on the random
attractor of the stochastically perturbed Lorenz
system governed by Eq. (1.13). This measure in-
dicates the probability of trajectories winding
up in a particular region of phase space and it is
very highly concentrated on the attractor, as in-
ferred from the huge range of density values: the
color bar in the figure is on a logarithmic scale,
and extends over more than 10 orders of mag-
nitude. The situation is thus very different from
that expected when studying additive noise —
in that case, the noise tends to smear out the
fine, Cantor-set–like structure of the determin-
istic, strange attractor and the associated PDF
has nonzero-volume support.
It hardly needs saying that additive noise has
been studied in climate dynamics much more
extensively than the multiplicative sort, for two
reasons: (i) it was easier to do so; and (ii) it was
suggested by the simple Brownian motion anal-
ogy of “weather ' water molecules” and “cli-
mate ' pollen particle,” as proposed by Has-
selmann (1976). Across the hierarchy of climate
models discussed in the previous two sections of
this article, however, it is clear that small-and-
fast scales of motion do not enter exclusively in
an additive manner: they pop up in many, if not
all terms of the governing equations, as summa-
rized in Eq. (1.12) above. The insights offered,
therefore, by Fig. 1.5 and the video are likely to
be of interest across the hierarchy of models, all
the way up to and including coupled GCMs and
Earth system models.
The invariant measure in Fig. 1.5 exhibits
amazing complexity, with fine, very intense fil-
amentation: there is no fuzziness whatsoever in
its topological structure, which does evoke the
Cantor-set foliation of the deterministic attrac-
tor. This fine structure strongly suggests that an
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12 M. Ghil
object of vanishing volume supports this mea-
sure, i.e. that the random attractor A(ω) of sys-
tem (1.12) has — like the strange attractor of
the classical, deterministic version, with σ = 0
— dimension smaller than 3.
1.5. A Rigorous Definition of
Climate Sensitivity
1.5.1. Equilibrium climate sensitivity
The usual view of climate sensitivity — as re-
flected in the work of the Intergovernmental
Panel on Climate Change [IPCC: (Houghton et
al., 1991; Solomon et al., 2007)] — is that of cli-
mate being in equilibrium, in the absence of ex-
ternal perturbations. In the setting of determin-
istic, autonomous dynamical systems, this view
can be described by the change in the position
of a fixed point, X0 = X0(µ), as a function of a
parameter µ.
Going back to Fig. 1.1, we can see how the
scalar functional T = T{X0(µ)}, namely the
global-mean temperature, varies as a function of
the fractional change µ of insolation at the top
of the atmosphere; here the fixed point X0 =
X0(µ) is the equilibrium solution T = T (x;µ)
of the EBM given by Eq. (1.2). Climate sensi-
tivity for the present climate is thus simply the
partial derivative ∂T /∂µ, i.e. the tangent of the
angle γ between the upper branch of Fig. 1.1
and the abscissa. The sensitivity increases, in
general, as we approach the bifurcation point
(X0, µ0) in Eqs. (1.3) or (1.4), and it decreases
away from it.
But we have seen in Secs. 1.3 and 1.4 here
that internal climate variability can be better
described by limit cycles and strange attractors
than by fixed points. Moreover, the presence
of time-dependent forcing, deterministic as well
as stochastic, introduces additional complexities
into the proper definition of climate sensitivity.
1.5.2. Defining climate sensitivity in
the presence of variability
We illustrate in Fig. 1.6 the difference between
the ways that a change in a parameter can affect
a climate model’s behavior in the case of purely
periodic solutions vs. the case of equilibrium so-
lutions. One might still think of the former case
as the climate of a simpler world, in which ENSO
would be purely periodic, rather than irregular.
In this case, climate sensitivity can no longer be
defined by a single scalar, like ∂T /∂µ, but needs
at least three scalars: the sensitivity of the mean
temperature along with that of the limit cycle’s
frequency (or period) and amplitude.
a) Equilibrium sensitivity
b) Nonequilibrium sensitivity
TCO2
t
t
t
T,CO2
T,CO2
T,CO2
Fig. 1.6. Climate sensitivity (a) for an equilibrium
model; and (b) for a nonequilibrium, oscillatory model.As a forcing (atmospheric CO2 concentration, say, dash-
dotted line) changes suddenly, global temperature (heavy
solid) undergoes a transition: in panel (a) only the meantemperature changes; in panel (b) the mean (shown now
as light dashed) adjusts as it does in panel (a), but the
amplitude of the oscillation can also decrease, increaseor stay the same.
More generally, the setting of non-
autonomous and of random dynamical systems,
as described in Sec. 1.4, allows one to examine
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A Mathematical Theory of Climate Sensitivity 13
the interaction of internal climate variability
with the forcing, whether natural or anthro-
pogenic, and to provide a definition of climate
sensitivity that takes into account the climate
systems non-equilibrium behavior and its time-
dependent forcing. Such a definition is of the
essence in studying systematically the sensitiv-
ity of global climate models (GCMs) to the un-
certainties in tens of semi-empirical parameters;
it will be given here in terms of the response of
the appropriate probability densities to changes
in the parameters. A comparison with numerical
results on parameter dependence for a somewhat
simplified GCM (Neelin et al., 2010) goes be-
yond the scope of this brief overview, and will
be given elsewhere.
As an illustration of the more general sensi-
tivity definition that we are proposing, we con-
sider here the case of an infinite-dimensional,
but still relatively simple ENSO model. The
model is due to Galanti and Tziperman (2000),
and its two dependent variables are sea surface
temperature T in the eastern Tropical Pacific
and thermocline depth h there, as a function of
time t:
T = f(T (t), h(t)), (1.14a)
h(t) = g(T, h, F )(t, τ1, τ2), (1.14b)
F (t) = 1 + εcos(ωt+ φ). (1.14c)
In Eqs. (1.14), F stands for the seasonal forc-
ing, with period 2π/ω = 12 months, and all
three variables — T, h and F — depend on the
time t and the delays τ1 and τ2; these delays
characterize the traveling times along the Equa-
tor of eastward Kelvin and westward Rossby
waves. Several authors have studied such delay-
differential models of ENSO; see Dijkstra (2005)
for a review and Ghil et al. (2008b) for fur-
ther mathematical details on this type of mod-
els. Note that this Galanti-Tziperman model is
non-autonomous, because of the seasonal forc-
ing, but it is still deterministic.
The solutions of Eqs. (1.14) exhibit periodic,
quasi-periodic and chaotic behavior, as well as
frequency locking to the time-dependent, sea-
sonal cycle. Thus, in principle, an infinite num-
ber of scalars are required to define the depen-
dence of these solutions on the parameters τ1and τ2; these scalars need to include not just
the means of temperature T and depth h, but
also their variance and higher-order moments.
We have chosen in Fig. 1.7 to represent this de-
pendence for the zeroth, second and fourth mo-
ments of h(t); more precisely the plotted quanti-
ties are the mean, standard deviation and fourth
root of the kurtosis.
0 1 2 3 4−40
−20
0
20
40
60
80
100
Change in % of τK,0=8.476
(<h>
−<
h>0)×
100
/<h>
0
Relative response in % of <h>
0 1 2 3 4−40
−30
−20
−10
0
10
Change in % of τK,0=8.476
(<h2 >1/
2 −<
h2 >1/2
0)×
100
/<h2 >1/
20
Relative response in % of <h2>1/2
0 1 2 3 4−100
−80
−60
−40
−20
0
20
Change in % of τK,0=8.476
(<h4 >1/
4 −<
h4 > 01/4 )×
100
/<h4 > 01/
4
Relative response in % of <h4>1/4
0 1 2 3 40
100
200
300
400
Change in % of τK,0=8.476
Rel
ativ
e re
spon
se in
%
Response in dW1
(<<h>>,<<h>>0
Fig. 1.7. Relative changes in the statistical properties
of the thermocline depth h(t), for the delay-differential
ENSO model of Eq. (1.14) and changes of 05% in the de-lay parameter τK associated with the Kelvin wave tran-
sit: (a) mean; (b) second moment; (c) fourth moment;
and (d) Wasserstein distance dW. Note intervals of bothsmooth and rough dependence of the solution on the pa-rameter. [Courtesy of M.D. Chekroun.]
The fourth quantity plotted in Fig. 1.7 is the
relative change in Wasserstein distance dW from
the same reference solution as for the other three
quantities. The Wasserstein distance or “earth
movers distance” dW is the distance between two
measures of equal mass on a metric space, i.e.,
on a space that has a metric attached to it, like
an n-dimensional Euclidean space (Dobrushin,
1970). Roughly speaking, dW represents the to-
tal work needed to move the “dirt” (i.e., the
measure) from a trench you are digging to
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14 M. Ghil
another one you are filling, over the distance be-
tween the two trenches. In general, the shape of
the two trenches and the depth along the trench
— i.e., the support of the measure and its den-
sity — can differ. In the case at hand, the shape
and density of the invariant measure that is be-
ing moved are plotted in Fig. 1.8.
−20
−10
0
10
20
30 −20 −15 −10 −5 0 5 10 15 20 25 30
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time-dependent invariant measure of the GT-model
h(t + 1)
h(t)
0.05
0.1
0.15
0.2
0.25
0.3
Fig. 1.8. Time-dependent invariant measure of the
Galanti and Tziperman (2000) model, plotted in an iso-
metric projection with the probability density on theperpendicular to the plane spanned by the coordinates
(h(t), h(t + 1)). The time here is in units of years, and
the density is highly concentrated on a very “thin” sup-port, as for the stochastically perturbed Lorenz model in
Fig. 1.5. [Courtesy of M.D. Chekroun.]
It is quite clear from Fig. 1.7 that intervals
of smooth dependence of the solution on the pa-
rameter alternate with jumps and with intervals
of rough dependence. The jump points are the
same in all four panels of the figure and there
is agreement also in the smooth vs. rough inter-
vals, although the Wasserstein distance in panel
(d) shows some roughness even over intervals in
which the three statistical moments in panels
(a–c) behave smoothly. The latter point is not
too surprising, since dW contains more informa-
tion than each of the moments.
The very high concentration of probability
density in the peak at the extreme left of Fig. 1.8
might seem surprising but actually agrees with
such a peak in the stochastically perturbed
ENSO model of Timmermann and Jin (2002).
The latter model is based on three ordinary
differential equations, for sea surface tempera-
tures in the eastern and western Tropical Pa-
cific and for thermocline depth, and it was an-
alyzed in detail by Chekroun et al. (2011b).
The corresponding near-singularity in invari-
ant measure on the model’s random attrac-
tor evolves regularly in time and thus suggests
that the interaction of deterministic nonlineari-
ties with time-dependent forcing, including even
stochastic perturbations, can help seasonal-to-
interannual prediction, rather than hinder it.
Chekroun et al. (2011a) have taken some inter-
esting steps in exploiting this possibility.
With the results illustrated in Figs. 1.7 and
1.8 in hand, it becomes natural to define climate
sensitivity in the presence of internal variabil-
ity and of time-dependent forcing as the partial
derivative of the Wasserstein distance dW with
respect to a parameter µ, ∂dW/∂µ. Clearly, dWhas to be defined in turn with respect to the
particular climate whose sensitivity we wish to
evaluate. As usual, it would be awkward, diffi-
cult or even impossible to compute with abso-
lute accuracy the function dW = dW(µ) and its
derivative at µ = µ0; but reasonable approxima-
tions should become available shortly, as is the
case already for the calculation of the invariant
measures of models of intermediate complexity.
The definition outlined here for a deterministic
but non-autonomous ENSO model can be gen-
eralized further to the random case, assuming
the existence of the suitable invariant measures.
A complementary approach to climate sensi-
tivity out of equilibrium is the one based on the
fluctuation-dissipation theory of statistical me-
chanics (Leith, 1975; Ghil et al., 1985; Lucarini
and Sarno, 2011). The complementarity arises
from D. Ruelle’s extending the many-particle,
statistical-mechanics ideas to low-order dynami-
cal systems subject to certain mathematical con-
ditions on the latter (Ruelle, 1997; Chekroun et
al., 2011b).
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A Mathematical Theory of Climate Sensitivity 15
We thus expect the theory of random dy-
namical systems to provide robust tools for
studying the parameter dependence of a nonlin-
ear, randomly perturbed system’s various “met-
rics.” These metrics — a word used by the cli-
mate community in a much broader sense than
the standard mathematical term — can include
global quantities, like mean temperature or to-
tal energy, but also much finer functionals of the
state of the system, such as regional tempera-
tures or precipitation. In addition, this theory
can help improve prediction of future system
properties, by relying on a judicious combina-
tion of the history of its slow and fast behavior.
1.6. Concluding Remarks
A complete theory of climate variability, across
the entire range of time scales of interest, is
still in the future. We have shown, though, that
powerful conceptual and numerical tools exist
in order to organize the emerging knowledge so
far. The approach described herein relies on ap-
plying systematically dynamical systems theory,
both deterministic and stochastic, across a hi-
erarchy of models, from the simplest toy mod-
els to the most detailed, coupled GCMs. This
approach has progressed from its first modest
steps, taken half-a-century ago, to the analysis
of the behavior of atmospheric, oceanic and cou-
pled GCMs over the last two decades. Particu-
larly interesting strides have been taken over the
last decade in studying the interaction of the
faster time scales with the slower ones, within a
genuinely nonlinear framework.
Acknowledgments
It is a pleasure to thank Michael D. Chekroun
for Figs.1.7 and 1.8. Discussions with him and
with several other colleagues including but not
restricted to D. Kondrashov, J. C. McWilliams,
J. D. Neelin, E. Simonnet, S. Wang, and I. Zali-
apin have helped develop the ideas in this chap-
ter, and more particularly those formulated in
Sec. 5. Related work is partially supported by
NSF grant DMS-0934426 and by by ONR-MURI
grant N00014-12-1-0911.
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